Sully is named after the airline pilot Chesley "Sully" Sullenberger, who landed a passenger jet on the Hudson River in 2009, saving all 155 passengers and crew on board.
Earlier Monday, Trump had posted that he was "Looking forward to being with the Bush Family to pay my respects to President George H.W. Bush".
The flag-draped casket of former President George H.W. Bush is carried by a joint services military honor guard to Special Air Mission 41 at Ellington Field during a departure ceremony Monday, Dec. 3, 2018, in Houston.
One of Bush's former senior aides, Ron Kaufman, says Bush's death marks "the end of a culture - a culture of civility". Trump and the first lady, Melania Trump, are scheduled to attend.
"A great joy to welcome home the newest member of our family, 'Sully, ' a handsome, and beautifully trained, lab from America's VetDogs".
A similar outpouring is anticipated in Washington this week during the state funeral for Bush, who died late Friday at his home in Houston.
In Texas, students, staff and visitors have been flocking to Bush's presidential library on the campus of Texas A&M University. "Could not be more grateful, especially for their commitment to our veterans", Bush said on Twitter about the dog in June.More news: Salman Khan tops Forbes India celebrity list, again
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House Speaker Paul Ryan, R-Wis., remembered volunteering for Bush's 1988 presidential campaign, including handing out literature at ballgames and at '"the Rock County 4-H fair". H.W. Bush will be honored at several public and private events in Houston and Washington before his burial on Thursday in Texas.
"I think he was one of the kindest, most generous men", said Marge Frazier, who visited the downtown statue on Sunday while showing friends from California around.
Bush's remains will lie in state at the U.S. Capitol Rotunda from Monday night until Wednesday.
Trump has ordered the federal government to close on Wednesday and both the New York Stock Exchange and Nasdaq will be closed in observance.
"Our thoughts and prayers are with the entire Bush family during this hard time", said John Miller, president and CEO of America's VetDogs.
Bush's passing puts him back in the Washington spotlight after more than two decades living the relatively low-key life of a former president.